Compounded Quarterly
We will calculate $100 growing to $200
Years = log(total/principal) / n*log(1 + rate/n)
For quarterly compounding n = 4
Years = log(200 / 100) / 4 * log (1 + .09/4)
Years = log(2) / 4 * log ( 1.0225 )
Years = 0.30102999566 / (4 * 0.0096633166794)
Years = 0.30102999566 / 0.0386532667
Years = 7.7879574318
Compounded Continuously
We will calculate $100 growing to $200
Years = natural log (total/principal) / rate
Years = natural log (200 / 100) / .09
Years = natural log (2) / .09
Years = 0.69314718056 / .09
Years = 7.7016353396